Dimension reduction numerical closure method for advection–diffusion-reaction systems

Many natural physical processes allow different mathematical descriptions on different scales. The microscale description is usually based on fundamental conservation laws that form a closed system of ordinary differential equations (ODEs) or partial differential equations (PDEs) but the numerical discretization of these equations may produce a system of ODEs with an enormous number of unknowns. Furthermore, time integration of the microscale equations usually requires time steps that are smaller than the observation time by many orders of magnitude. A direct solution of these ODEs can be extremely expensive. Often, we are only interested in the average behavior of the microscale system rather than the exact solution of the ODEs. Here we propose a novel dimension reduction computational closure (DRNC) method that gives an approximate solution of the ODEs and provides an accurate prediction of the average behavior. The DRNC method consists of two main elements. First, effective ODEs for evolution of average variables (e.g. average velocity, concentration and mass of a mineral precipitate) are obtained by averaging the micro-scale ODEs over the entire micro-scale domain. These effective ODEs contain non-local terms in the form of volume integrals of functions of the micro-scale variables. Second, a numerical closure is used to close the system of the effective equations. The numerical closure is achieved via short bursts of the microscale model. The DRNC method is used to simulate flow and transport with mixing controlled reactions and mineral precipitation by reducing porescale (microscale) Navier–Stokes and advection–diffusion-reaction equations to ODEs for averaged velocity and concentrations. Good agreement between direct solutions of the microscale equations and DRNC solutions for different boundary conditions and Damkohler numbers confirms the accuracy and computational efficiency of DRNC method. The DRNC method significantly accelerates microscale simulations, while providing accurate approximation of the microscale solution and accurate prediction of the average behavior of the system.